Practically, it works as a kind of "approximated oracle": soft-labels are assigned to latent states over the trajectories by averaging over the posterior, updated using the parametrization obtained from the previous maximization phase, starting from initial guesses[1]}.
The algorithm iterates until convergence, tested by stopping conditions on the likelihood or maximum number of iterations.
As common for EM, several initialization are required, by sampling different values of the parameters to tackle eventual convergence to poor local minima. Notably, EM has interesting convergence properties, holding also when a closed form solution is not available for the M-step [2]}. In this case, an increase of the \(\mathcal {Q}\) -function is required in each iteration, leading to the Generalized EM algorithm (GEM).
